In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site.
This can lead to complicated band structures because the orbitals belong to different point-group representations.
The reciprocal lattice and the Brillouin zone often belong to a different space group than the crystal of the solid.
So the tight-binding model can provide nice examples for those who want to learn more about group theory.
The tight-binding model has a long history and has been applied in many ways and with many different purposes and different outcomes.
[2] In the study of conductive polymers, organic semiconductors and molecular electronics, for example, tight-binding-like models are applied in which the role of the atoms in the original concept is replaced by the molecular orbitals of conjugated systems and where the interatomic matrix elements are replaced by inter- or intramolecular hopping and tunneling parameters.
By 1928, the idea of a molecular orbital had been advanced by Robert Mulliken, who was influenced considerably by the work of Friedrich Hund.
The LCAO method for approximating molecular orbitals was introduced in 1928 by B. N. Finklestein and G. E. Horowitz, while the LCAO method for solids was developed by Felix Bloch, as part of his doctoral dissertation in 1928, concurrently with and independent of the LCAO-MO approach.
A much simpler interpolation scheme for approximating the electronic band structure, especially for the d-bands of transition metals, is the parameterized tight-binding method conceived in 1954 by John Clarke Slater and George Fred Koster,[1] sometimes referred to as the SK tight-binding method.
With the SK tight-binding method, electronic band structure calculations on a solid need not be carried out with full rigor as in the original Bloch's theorem but, rather, first-principles calculations are carried out only at high-symmetry points and the band structure is interpolated over the remainder of the Brillouin zone between these points.
In the recent research about strongly correlated material the tight binding approach is basic approximation because highly localized electrons like 3-d transition metal electrons sometimes display strongly correlated behaviors.
However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.
In 2019, Bannwarth et al. introduced the GFN2-xTB method, primarily for the calculation of structures and non-covalent interaction energies.
to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals
The Bloch theorem states that the wave function in a crystal can change under translation only by a phase factor: where
It is also called the bond energy or two center integral and it is the dominant term in the tight binding model.
These, too, are typically small; if not, then Pauli repulsion has a non-negligible influence on the energy of the central atom.
The interatomic distances can be too small or the charges on the atoms or ions in the lattice is wrong for example.
Energies and eigenstates on some high symmetry points in the Brillouin zone can be evaluated and values integrals in the matrix elements can be matched with band structure data from other sources.
If they are large it is again an indication that the tight binding model is of limited value for some purposes.
Broad bands in dense materials are better described by a nearly free electron model.
The model also gives good results in the case of open crystal structures, like diamond or silicon, where the number of neighbors is small.
[2] Bloch functions describe the electronic states in a periodic crystal lattice.
are called Wannier functions, and are fairly closely localized to the atomic site
Using the atomic orbital as a basis state, the second quantization Hamiltonian operator in the tight binding framework can be written as: Here, hopping integral
To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals where N = total number of sites and
can be represented in the familiar form of the energy dispersion: This example is readily extended to three dimensions, for example, to a body-centered cubic or face-centered cubic lattice by introducing the nearest neighbor vector locations in place of simply n a.
In 1954 J.C. Slater and G.F. Koster published, mainly for the calculation of transition metal d-bands, a table of interatomic matrix elements[1] which can also be derived from the cubic harmonic orbitals straightforwardly.
The table expresses the matrix elements as functions of LCAO two-centre bond integrals between two cubic harmonic orbitals, i and j, on adjacent atoms.
for sigma, pi and delta bonds (Notice that these integrals should also depend on the distance between the atoms, i.e. are a function of